Several lines of observational evidence suggest that white dwarfs receive small birth kicks due to anisotropic mass loss. If other stars possess extrasolar analogues to the Solar Oort cloud, the orbits of comets in such clouds will be scrambled by white dwarf natal kicks. Although most comets will be unbound, some will be placed on low angular momentum orbits vulnerable to sublimation or tidal disruption. The dusty debris from these comets will manifest itself as a debris disk temporarily visible around newborn white dwarfs; examples of such disks may already have been seen in the Helix Nebula, and around several other young WDs. Future observations with the James Webb Space Telescope may distinguish this hypothesis from alternatives such as a dynamically excited Kuiper Belt analogue. If interpreted as indeed being cometary in origin, the observation that >15% of young WDs possess such disks provides indirect evidence that low mass gas giants (thought necessary to produce an Oort cloud) are common in the outer regions of extrasolar planetary systems. Hydrogen abundances in the atmospheres of older white dwarfs can, if sufficiently low, also be used to place constraints on the joint parameter space of natal kicks and exo-Oort cloud models.